imasf1oxmet - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > imasf1oxmet		Structured version Visualization version GIF version

Theorem imasf1oxmet 24325

Description: The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
imasf1oxmet.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasf1oxmet.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasf1oxmet.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
imasf1oxmet.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasf1oxmet.e	⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
imasf1oxmet.d	⊢ 𝐷 = (dist‘𝑈)
imasf1oxmet.m	⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))

**Assertion**
Ref	Expression
imasf1oxmet	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Proof of Theorem imasf1oxmet Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasf1oxmet.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasf1oxmet.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasf1oxmet.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
4		f1ofo 6845	. . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
6		imasf1oxmet.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
7		eqid 2725	. . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅)
8		imasf1oxmet.d	. . . 4 ⊢ 𝐷 = (dist‘𝑈)
9	1, 2, 5, 6, 7, 8	imasdsfn 17499	. . 3 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
10	1	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑈 = (𝐹 “_s 𝑅))
11	2	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
12	3	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐹:𝑉–1-1-onto→𝐵)
13	6	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑅 ∈ 𝑍)
14		imasf1oxmet.e	. . . . . . . 8 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
15		imasf1oxmet.m	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))
16	15	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐸 ∈ (∞Met‘𝑉))
17		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
18		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
19	10, 11, 12, 13, 14, 8, 16, 17, 18	imasdsf1o 24324	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
20		xmetcl 24281	. . . . . . . . 9 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎𝐸𝑏) ∈ ℝ^*)
21	20	3expb 1117	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
22	15, 21	sylan 578	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
23	19, 22	eqeltrd 2825	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
24	23	ralrimivva 3190	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
25		f1ofn 6839	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉)
26	3, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
27		oveq2 7427	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎)𝐷𝑦) = ((𝐹‘𝑎)𝐷(𝐹‘𝑏)))
28	27	eleq1d 2810	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
29	28	ralrn 7097	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
30	26, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
31		forn 6813	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
32	5, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
33	32	raleqdv 3314	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
34	30, 33	bitr3d 280	. . . . . 6 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
35	34	ralbidv 3167	. . . . 5 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
36	24, 35	mpbid 231	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*)
37		oveq1 7426	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥𝐷𝑦) = ((𝐹‘𝑎)𝐷𝑦))
38	37	eleq1d 2810	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
39	38	ralbidv 3167	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
40	39	ralrn 7097	. . . . . 6 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
41	26, 40	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
42	32	raleqdv 3314	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
43	41, 42	bitr3d 280	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
44	36, 43	mpbid 231	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*)
45		ffnov 7547	. . 3 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ↔ (𝐷 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
46	9, 44, 45	sylanbrc 581	. 2 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶ℝ^*)
47		xmeteq0 24288	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
48	16, 17, 18, 47	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
49	19	eqeq1d 2727	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝑎𝐸𝑏) = 0))
50		f1of1 6837	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–1-1→𝐵)
51	3, 50	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝐵)
52		f1fveq 7272	. . . . . . . 8 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
53	51, 52	sylan 578	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
54	48, 49, 53	3bitr4d 310	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
55	16	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐸 ∈ (∞Met‘𝑉))
56		simpr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑐 ∈ 𝑉)
57	17	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑎 ∈ 𝑉)
58	18	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑏 ∈ 𝑉)
59		xmettri2 24290	. . . . . . . . . 10 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑐 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
60	55, 56, 57, 58, 59	syl13anc 1369	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
61	19	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
62	10	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑈 = (𝐹 “_s 𝑅))
63	11	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
64	12	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐹:𝑉–1-1-onto→𝐵)
65	13	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑅 ∈ 𝑍)
66	62, 63, 64, 65, 14, 8, 55, 56, 57	imasdsf1o 24324	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑎)) = (𝑐𝐸𝑎))
67	62, 63, 64, 65, 14, 8, 55, 56, 58	imasdsf1o 24324	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑏)) = (𝑐𝐸𝑏))
68	66, 67	oveq12d 7437	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) = ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
69	60, 61, 68	3brtr4d 5181	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
70	69	ralrimiva 3135	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
71		oveq1 7426	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑎)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑎)))
72		oveq1 7426	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑏)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))
73	71, 72	oveq12d 7437	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑐) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) = (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
74	73	breq2d 5161	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
75	74	ralrn 7097	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
76	26, 75	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
77	32	raleqdv 3314	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
78	76, 77	bitr3d 280	. . . . . . . 8 ⊢ (𝜑 → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
79	78	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
80	70, 79	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
81	54, 80	jca 510	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
82	81	ralrimivva 3190	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
83	27	eqeq1d 2727	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0))
84		eqeq2 2737	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎) = 𝑦 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
85	83, 84	bibi12d 344	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ↔ (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏))))
86		oveq2 7427	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑏) → (𝑧𝐷𝑦) = (𝑧𝐷(𝐹‘𝑏)))
87	86	oveq2d 7435	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
88	27, 87	breq12d 5162	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
89	88	ralbidv 3167	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
90	85, 89	anbi12d 630	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑏) → (((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
91	90	ralrn 7097	. . . . . . 7 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
92	26, 91	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
93	32	raleqdv 3314	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
94	92, 93	bitr3d 280	. . . . 5 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
95	94	ralbidv 3167	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
96	82, 95	mpbid 231	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
97	37	eqeq1d 2727	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷𝑦) = 0))
98		eqeq1 2729	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥 = 𝑦 ↔ (𝐹‘𝑎) = 𝑦))
99	97, 98	bibi12d 344	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦)))
100		oveq2 7427	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑧𝐷𝑥) = (𝑧𝐷(𝐹‘𝑎)))
101	100	oveq1d 7434	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))
102	37, 101	breq12d 5162	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
103	102	ralbidv 3167	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
104	99, 103	anbi12d 630	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
105	104	ralbidv 3167	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
106	105	ralrn 7097	. . . . 5 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
107	26, 106	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
108	32	raleqdv 3314	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
109	107, 108	bitr3d 280	. . 3 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
110	96, 109	mpbid 231	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
111	15	elfvexd 6935	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
112		focdmex 7960	. . . 4 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
113	111, 5, 112	sylc 65	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
114		isxmet 24274	. . 3 ⊢ (𝐵 ∈ V → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
115	113, 114	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
116	46, 110, 115	mpbir2and 711	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 class class class wbr 5149 × cxp 5676 ran crn 5679 ↾ cres 5680 Fn wfn 6544 ⟶wf 6545 –1-1→wf1 6546 –onto→wfo 6547 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 0cc0 11140 ℝ^*cxr 11279 ≤ cle 11281 +_𝑒 cxad 13125 Basecbs 17183 distcds 17245 “_s cimas 17489 ∞Metcxmet 21281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-0g 17426 df-gsum 17427 df-xrs 17487 df-imas 17493 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-xmet 21289

This theorem is referenced by: imasf1omet 24326 xpsxmet 24330 imasf1obl 24441 imasf1oxms 24442

W3C validator